Burchnall-Chaundy theory for skew Poincar\'e-Birkhoff-Witt extensions

TL;DR

This paper extends classical algebraic dependence results for commuting elements to the broader context of skew Poincaré-Birkhoff-Witt extensions, enriching the understanding of noncommutative algebra structures.

Contribution

It generalizes Burchnall-Chaundy theory to skew Poincaré-Birkhoff-Witt extensions, expanding its applicability in noncommutative algebra.

Findings

Classical results are extended to a new algebraic setting.

The paper establishes algebraic dependence properties in skew PBW extensions.

It provides a framework for analyzing commuting elements in complex noncommutative rings.

Abstract

In this paper we review some classical results on the algebraic dependence of commuting elements in several noncommutative algebras as differential operator rings and Ore extensions. Then we extend all these results to a more general setting, the family of noncommutative rings known as skew Poincar\'e-Birkhoff-Witt extensions.